"Writing 1 in form of 1t1+⋯+1tn" was answered by our community

Charity Odom

Answered question

2022-03-07

Writing 1 in form of

\frac{1}{t_{1}} + \dots + \frac{1}{t_{n}}

stadfeste8ru

Beginner2022-03-08Added 1 answers

Hint.

\frac{1}{n} + \frac{1}{n} = \frac{1}{n} + \frac{1}{n + 1} + \frac{1}{n (n + 1)} .

For example, say

N = 3

. Then we can write:

1 = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}

= \frac{1}{4} + \frac{1}{5} + \frac{1}{20} + \frac{1}{5} + \frac{1}{20} + \frac{1}{5} + \frac{1}{20}

= \frac{1}{4} + \frac{1}{5} + \frac{1}{20} + \frac{1}{6} + \frac{1}{30} + \frac{1}{21} + \frac{1}{420} + \frac{1}{6} + \frac{1}{30} + \frac{1}{21} + \frac{1}{420}

= \frac{1}{4} + \frac{1}{5} + \frac{1}{20} + \frac{1}{6} + \frac{1}{30} + \frac{1}{21} + \frac{1}{420} + \frac{1}{7} + \frac{1}{42} + \frac{1}{31}

{+} \frac{1}{930} + \frac{1}{22} + \frac{1}{462} + \frac{1}{421} + \frac{1}{176820}

= \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{20} + \frac{1}{21} + \frac{1}{22} + \frac{1}{30} + \frac{1}{31} + \frac{1}{42}

{+} \frac{1}{420} + \frac{1}{421} + \frac{1}{462} + \frac{1}{930} + \frac{1}{176820} .

Keon Moore

Beginner2022-03-09Added 1 answers

Another possibility is to use Fibonacci's Greedy algorithm.
Start with

t_{1} = M + 1

and then proceed inductively: define

t_{n + 1}

to be the smallest natural number greater than t_n for which

\frac{1}{t_{1}} + \frac{1}{t_{2}} + \dots + \frac{1}{t_{n + 1}} \leq 1 .

In other words:

t_{1} = M + 1

t_{n + 1} = m \in N ∣ m > t_{n} and \frac{1}{t_{1}} + \frac{1}{t_{2}} + \dots + \frac{1}{t_{n}} + \frac{1}{m} \leq 1 =

= max ⌈ {(1 - \frac{1}{t_{1}} - \frac{1}{t_{2}} - \dots - \frac{1}{t_{n}})}^{- 1} ⌉, t_{n} + 1

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